Corresponding squares (also called relative squares, sister squares and coordinate squares ) in chess

Chess

Chess is a two-player board game played on a chessboard, a square-checkered board with 64 squares arranged in an eight-by-eight grid. It is one of the world's most popular games, played by millions of people worldwide at home, in clubs, online, by correspondence, and in tournaments.Each player...

 occur in some chess endgames, usually ones that are mostly blocked. If squares x and y are corresponding squares, it means that if one player moves to x then the other player must move to y in order to hold his position. Usually there are several pairs of these squares, and the members of each pair are labeled with the same number, e.g. 1, 2, etc. In some cases they indicate which square the defending king

King (chess)

In chess, the king is the most important piece. The object of the game is to trap the opponent's king so that its escape is not possible . If a player's king is threatened with capture, it is said to be in check, and the player must remove the threat of capture on the next move. If this cannot be...

 must move to in order to keep the opposing king away. In other cases, a maneuver by one king puts the other player in a situation where he cannot move to the corresponding square, thus the first king is able to penetrate the position . The theory of corresponding squares is more general than opposition

Opposition (chess)

In chess, opposition is the situation occurring when two kings face each other on a rank or file, with only one square in between them. In such a situation, the player not having to move is said to "have the opposition" . It is a special type of zugzwang and most often occurs in endgames with...

, and is more useful in cluttered positions.

Details

Corresponding squares are squares of reciprocal (or mutual) zugzwang

Zugzwang

Zugzwang is a term usually used in chess which also applies to various other games. The term finds its formal definition in combinatorial game theory, and it describes a situation where one player is put at a disadvantage because he has to make a move when he would prefer to pass and make no move...

. They occur most often in king and pawn endgames, especially with triangulation

Triangulation (chess)

Triangulation is a tactic used in chess to put one's opponent in zugzwang. That is, it gets to the same position with the other player to move, when it is a disadvantage for that player to move, e.g. he has to give up a blockade and let the other player penetrate his position...

, opposition

Opposition (chess)

In chess, opposition is the situation occurring when two kings face each other on a rank or file, with only one square in between them. In such a situation, the player not having to move is said to "have the opposition" . It is a special type of zugzwang and most often occurs in endgames with...

, and mined squares (see Zugzwang#Mined squares). A square that White can move to corresponds to a square that Black can move to. If one player moves to such a square, the opponent moves to the corresponding square to put the opponent in zugzwang .

A simple example

One of the simplest and most important uses of corresponding squares is in this king and pawn versus king endgame. Assume that the black king is in front of the pawn and the white king is behind or to the side of the pawn. The black king is trying to block the white pawn and the white king is supporting its pawn. If the white king gets to any of the key square

Key square

In chess, particularly in endgames, a key square is a square such that if a player's king can occupy it, he can force some gain such as the promotion of a pawn or the capture of an opponent's pawn. Key squares are useful mostly in endgames involving only kings and pawns...

s (marked with "x"), he wins. Suppose the black king moves to the square labeled "1" near him (square c8). Then if the white king moves to the corresponding square (also labeled "1", square c6), he wins. Conversely, if the white king moves to the "1" square then the black king must move to the corresponding square to draw

Draw (chess)

In chess, a draw is when a game ends in a tie. It is one of the possible outcomes of a game, along with a win for White and a win for Black . Usually, in tournaments a draw is worth a half point to each player, while a win is worth one point to the victor and none to the loser.For the most part,...

. Thus if both kings are on the "1" squares, the position is a reciprocal zugzwang. Note that the second player moving to one of the corresponding squares has the advantage. Being on a square when the opponent is not on the corresponding square is a disadvantage.

The squares labeled "2" are similar corresponding squares. If the white king is on the d5 square (the middle one labeled "3"), he is threatening to move to either the "1" square or the "2" square. Therefore the black king must be in a position to move to either his "1" square or his "2" square in order to hold the draw, so he must be on one of his "3" squares. This makes the defense for Black clear: shift between the squares labeled "3" until the white king moves to his "1" or "2" square, and then go to the corresponding square, gaining the opposition. If the black king moves to the "1" or "2" squares under any other circumstances, the white king moves to the corresponding square, takes the opposition, the black king moves, and White advances the pawn and will promote

Promotion (chess)

Promotion is a chess rule describing the transformation of a pawn that reaches its eighth rank into the player's choice of a queen, knight, rook, or bishop of the same color . The new piece replaces the pawn on the same square and is part of the move. Promotion is not limited to pieces that have...

 it and win, with a basic checkmate

Checkmate

Checkmate is a situation in chess in which one player's king is threatened with capture and there is no way to meet that threat. Or, simply put, the king is under direct attack and cannot avoid being captured...

.

The c5 and e5 squares can also be label "3" squares, since if the white king is on one of them, the black king must be on one of his "3" squares to draw.

A second example

This is another example that is fairly simple. The key squares (see king and pawn versus king endgame) are e1, e2, e3, and f3. If the black king gets to any of those squares, Black wins. The job of the white king is to keep the black king off those squares. One might think that Black has the advantage, since he has the opposition

Opposition (chess)

In chess, opposition is the situation occurring when two kings face each other on a rank or file, with only one square in between them. In such a situation, the player not having to move is said to "have the opposition" . It is a special type of zugzwang and most often occurs in endgames with...

. White can defend the two key squares of e3 and f3 by oscillating between e2 and f2. White's defense is simple if he observes the corresponding squares:

1. Kf2! (keeping the black king off e3 and f3)

1... Kd3

2. Kf3! moving to the corresponding square

2... Kd2

3. Kf2! Kd1

4. Kf1!

Each time the black king moves to a numbered square, the white king moves to the corresponding square .

An example with separated key squares

In this position, the squares marked with "x" are key squares and the e1 square is a "5" for White. If White occupies any of the key squares, he wins. With separated key squares, the shortest path connecting them is significant. If White is to move in this position, he wins by seizing a key square by moving to e2 or f2. If Black is to move, he draws by moving to his "5" square. Black maintains the draw by always moving to the square corresponding to the one occupied by the white king .

An example with triangulation

In this position, e2, e3, and d4 are key squares. If the white king can reach any of them, White wins. The black king cannot move out of the "square" of White's d-pawn (see king and pawn versus king endgame), otherwise it will promote

Promotion (chess)

Promotion is a chess rule describing the transformation of a pawn that reaches its eighth rank into the player's choice of a queen, knight, rook, or bishop of the same color . The new piece replaces the pawn on the same square and is part of the move. Promotion is not limited to pieces that have...

. The square c3 is adjacent to d4 and the "1" square the White king is on, so it is numbered "2". Therefore e3 is "2" for Black. White threatens to move to c2, so this is labeled "3". Since Black must be able to move to "1" and "2", f4 is his corresponding "3" square. If the White king is on b2 or b3, he is threatening to move to "2" or to "3", so those are also "1" squares for him. White has more corresponding squares, so he can out-maneuver Black to win .

1. Kc2 Kf4

2. Kb3 Kf3

3. Kb2 Kf4 The black king must leave his "1" square, and has no corresponding "1" square to which to move.

4. Kc2! Kf3 The white king moved to his "3" square but the black king is on his "3" square, so he cannot move to "3". White has used triangulation

Triangulation (chess)

Triangulation is a tactic used in chess to put one's opponent in zugzwang. That is, it gets to the same position with the other player to move, when it is a disadvantage for that player to move, e.g. he has to give up a blockade and let the other player penetrate his position...

.

5. Kd2 Back to the starting position, but with Black to move.

5... Kf4 Black is on his "1" square, so cannot move to a "1" square.

6. Ke2!

White occupies a key square and can support the advance of his pawn until he is able to win the black pawn, e.g.: 6... Kf5 7. Ke3 Ke5 8. d4+ Kd5 9. Kd3 Kd6 10. Ke4 Ke6 11. d5+ Kd6 12. Kd4 Kd7 13. Kc5.

Lasker-Reichhelm position

One of the most famous and complicated positions solved with the method of corresponding squares is this endgame study

Endgame study

An endgame study, or just study, is a composed chess position—that is, one that has been made up rather than one from an actual game—presented as a sort of puzzle, in which the aim of the solver is to find a way for one side to win or draw, as stipulated, against any moves the other side...

 composed by World Champion Emanuel Lasker

Emanuel Lasker

Emanuel Lasker was a German chess player, mathematician, and philosopher who was World Chess Champion for 27 years...

 and Gustavus Charles Reichhelm in 1901. It is described in the 1932 treatise L'opposition et cases conjuguées sont réconciliées (Opposition and Sister Squares are Reconciled), by Vitaly Halberstadt

Vitaly Halberstadt

Vitaly Halberstadt was a French chess player, problemist and, above all, a noted endgame study composer.Born in Odessa, Russian Empire , he emigrated to France...

 and Marcel Duchamp

Marcel Duchamp

Marcel Duchamp was a French artist whose work is most often associated with the Dadaist and Surrealist movements. Considered by some to be one of the most important artists of the 20th century, Duchamp's output influenced the development of post-World War I Western art...

.

1. Kb1 Kb7

2. Kc1 Kc7

3. Kd1 Kd8

4. Kc2 Kc8

5. Kd2 Kd7

6. Kc3 Kc7

7. Kd3 Kb6

8. Ke3

and White wins by penetrating on the kingside. Each of White's first seven moves are the only one that wins .

See also

• Chess endgame
• King and pawn versus king endgame
• Key square
  Key square
  In chess, particularly in endgames, a key square is a square such that if a player's king can occupy it, he can force some gain such as the promotion of a pawn or the capture of an opponent's pawn. Key squares are useful mostly in endgames involving only kings and pawns...
  
  
• Opposition
  Opposition (chess)
  In chess, opposition is the situation occurring when two kings face each other on a rank or file, with only one square in between them. In such a situation, the player not having to move is said to "have the opposition" . It is a special type of zugzwang and most often occurs in endgames with...
  
  
• Triangulation
  Triangulation (chess)
  Triangulation is a tactic used in chess to put one's opponent in zugzwang. That is, it gets to the same position with the other player to move, when it is a disadvantage for that player to move, e.g. he has to give up a blockade and let the other player penetrate his position...
  
  
• Zugzwang
  Zugzwang
  Zugzwang is a term usually used in chess which also applies to various other games. The term finds its formal definition in combinatorial game theory, and it describes a situation where one player is put at a disadvantage because he has to make a move when he would prefer to pass and make no move...